CoCalc -- Capstone-1(week9).ipynb

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Kernel: SageMath 9.7

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#Capstone Week 9 Code

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#Lokta-Volterra (Predator Prey Model)

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#Defining the differential equations:

#T' = bT - BST
#S' = mBST - dS

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#Defining the differential equations: *added time delay*

#T' = b(T-2) - BS(T-2)
#S' = mBS(T-2) - dS

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#First we will be using set values for the parameters (B, S, m, b), and manipulating the equation to see the affects of time delay.

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#To compare the effect of each change, we will be plotting the time series and trajectory for each set of ics

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#setting b=0.6, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [0.6*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] 

# initial conditions
ic = [40, 80] # T=40, S=80

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting trajectory for ics T=40, S=80
list_plot(sol, axes_labels=['N','P'])

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#setting b=5, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S] 

# setting the first pair of initial conditions
ic = [40, 80] # T=40, S=80

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting the time series for ics T=40, S=80 and change in beta=5

sol = desolve_odeint([5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S] , dvars=[T,S], ics=ic, times=t)

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# plotting trajectory for ics T=40, S=80, B= 5
list_plot(sol, axes_labels=['N','P'])

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#setting b=0.6, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S]  

# setting the second pair of initial conditions
ic = [100, 200] # T=100, S=200

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting the time series for ics T=100, S=200

sol = desolve_odeint([5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S] , dvars=[T,S], ics=ic, times=t)

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# plotting trajectory for ics T=100, S=200
list_plot(sol, axes_labels=['N','P'])

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#setting b=5, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [5*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] 

# setting the second pair of initial conditions
ic = [100, 200] # T=100, S=200

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting the time series for ics T=100, S=200

sol = desolve_odeint([5*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] , dvars=[T,S], ics=ic, times=t)

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# plotting trajectory for ics T=100, S=200, b= 5
list_plot(sol, axes_labels=['N','P'])

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#setting b=0.6, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S] 

# setting the third pair of initial conditions
ic = [5, 50] # T=5, S=50

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting the time series for ics T=5, S=50

sol = desolve_odeint([5*(T-2) - 0.02*(T-2)*S, 0.6*0.02*(T-2)*S - .3*S] , dvars=[T,S], ics=ic, times=t)

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# plotting trajectory for ics T=5, S=50
list_plot(sol, axes_labels=['N','P'])

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#setting b=5, m=0.6, S=0.02, B= 0.3

# Define the differential equation
var('T','S')
DEs = [5*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] 

# setting the third pair of initial conditions
ic = [5, 50] # T=5, S=50

# setting the timespan
t = srange(0,100,.1) 

sol = desolve_odeint(DEs, dvars=[T,S], ics=ic, times=t)

# plotting the time series for ics T=5, S=50

sol = desolve_odeint([5*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] , dvars=[T,S], ics=ic, times=t)

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# plotting trajectory for ics T=5, S=50, b=5
list_plot(sol, axes_labels=['N','P'])

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---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File /ext/sage/9.7/src/sage/plot/plot.py:3088, in list_plot(data, plotjoined, **kwargs)
   3087 from sage.rings.real_double import RDF
-> 3088 RDF(data[0])
   3089 data = list(enumerate(data))
TypeError: 'zip' object is not subscriptable

During handling of the above exception, another exception occurred:
TypeError                                 Traceback (most recent call last)
Input In [2], in <cell line: 8>()
      4     sol=desolve_odeint([birthrate*N - RealNumber('0.01')*N*P, RealNumber('0.5')*RealNumber('0.01')*N*P - RealNumber('0.2')*P], ics=[Integer(100),Integer(200)],
      5     dvars=[N,P], times=t)
      6     return list_plot(zip(t,sol[:,Integer(1)]), legend_label ="N") + list_plot(zip(t,sol[:,Integer(0)]), color = "red", legend_label = "P")
----> 8 laziness(RealNumber('0.7'))
      9 laziness(RealNumber('0.9'))
     10 laziness(RealNumber('1.1'))
Input In [2], in laziness(birthrate)
      3 t = srange(Integer(0),Integer(100),RealNumber('0.1'))
      4 sol=desolve_odeint([birthrate*N - RealNumber('0.01')*N*P, RealNumber('0.5')*RealNumber('0.01')*N*P - RealNumber('0.2')*P], ics=[Integer(100),Integer(200)],
      5 dvars=[N,P], times=t)
----> 6 return list_plot(zip(t,sol[:,Integer(1)]), legend_label ="N") + list_plot(zip(t,sol[:,Integer(0)]), color = "red", legend_label = "P")
File /ext/sage/9.7/src/sage/misc/decorators.py:491, in options.__call__.<locals>.wrapper(*args, **kwds)
    489     options['__original_opts'] = kwds
    490 options.update(kwds)
--> 491 return func(*args, **options)
File /ext/sage/9.7/src/sage/plot/plot.py:3098, in list_plot(data, plotjoined, **kwargs)
   3089     data = list(enumerate(data))
   3090 except TypeError: # we can get this TypeError if the element is a list
   3091                   # or tuple or numpy array, or an element of CC, CDF
   3092     # We also want to avoid doing CC(data[0]) here since it will go
   (...)
   3096     # So, the only other check we need to do is whether data[0] is an
   3097     # element of the Symbolic Ring.
-> 3098     if data[0] in sage.symbolic.ring.SR:
   3099         data = list(enumerate(data))
   3101 try:
TypeError: 'zip' object is not subscriptable

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#Bifurcation diagram/Parameter Scan for LV differential equations:
#T' = b(T-2) - BS(T-2)
#S' = mBS(T-2) - dS

var('T','S')
t = srange(0,100,.1) #timespan
sol = desolve_odeint([5*T - 0.02*T*S, 0.6*0.02*T*S - .3*S] , dvars=[T,S], ics=ic, times=t)
list_plot(sol, plotjoined=true, axes_labels=["Time", "Population Density"])

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